Showing papers in "Algebra Universalis in 1981"

Dimension and automorphism groups of lattices

Abstract: Every group is the automorphism group of a lattice of order dimension at most 4 We conjecture that the automorphism groups of finite modular lattices of bounded dimension do not represent every finite group It is shown that ifp is a large prime dividing the order of the automorphism group of a finite modular latticeL then eitherL has high order dimension orMp, the lattice of height 2 and orderp+2, has a cover-preserving embedding inL We mention a number of open problems

On some identities valid in modular congruence varieties

Abstract: Freese and J6nsson [8] showed that the congruence lattice of a (universal) algebra in a congruence modular variety is always arguesian. On the other hand J6nsson [16] constructed arguesian lattices which cannot be embedded into the normal subgroup lattice of a group. These lattices consist of two arguesian planes of different prime order glued together over a two element sublattice (cf. Dilworth and Hall [3]). In [11], Herrmann and Poguntke derived identities not valid in those lattices but valid in all lattices of normal subgroups. In the present paper we show that these (and more general) identities hold in the congruence lattice of any algebra in a congruence modular variety. This implies, in particular, that the class of arguesian tattices does not form a congruence variety in the sense of J6nsson [17]. (This result has been proved by the first author and announced in [7]). Moreover, one concludes as in [11] that a modular congruence variety cannot be defined by finitely many identities provided it contains the rational projective plane or two projective planes of distinct prime orders or a subgroup lattice of a group C~:. 1. Definitions and main results For subgroups the verification of the lattice identities to be constructed reduces to the trivial observation that isomorphic abelian quotients have the same exponent. Consequently, we introduce \"projective\" lattice relations which yield for certain quotients: I isomorphy II \"coordinate systems\" allowing one to speak about \"exponents\". Ad L Projective quotients. If a and b are elements of a modular lattice such that a>-b then we write a/b={xl a>-x>-b}. We write a/b/~c/d as well as

Structural diversity in the lattice of equational theories

Abstract: This paper is principally concerned with conditions under which various partition lattices are isomorphic to intervals in either the lattice of equational theories extending a given equational theory or the lattice of subtheories of a given equational theory.

Sets of Natural Numbers of Positive Density and Cylindric Set Algebras of Dimension 2

Abstract: Cylindric algebras of higher dimensions are defined analogously (see [7, p . 164]) . Examples of cylindric algebras of sets are the projective algebras of subsets of the plane: classes of sets situated in the Euclidean plane, closed under Boolean operations and under projection onto either axis, and containing the direct product A X B whenever A and B are situated on the x-axis and y-axis respectively . In [11, p. 12], Ulam has asked some fundamental questions about projective algebras of sets in the plane (and higher dimensional Euclidean spaces) . It is the purpose of this paper to settle some of these questions . In §1 certain questions concerning sets of natural numbers of positive density are discussed which arise from computations involved in the construction in §2 of a countable collection of sets in the plane which is not contained in a finitely generated cylindric algebra of sets in the plane . In §3 we summarize the status of each of the other problems on projective algebras mentioned by Ulam in [11] . The first author is responsible for §1 while the results in §2 and §3 are due to the other two authors . We use the following notation and terminology . Each ordinal is identified with the set of ordinals smaller than it . Each initial ordinal is identified with its cardinality. The first infinite ordinal is w . Often we call a cylindric algebra of sets, a cylindric set algebra . We should also point out that our notation and terminology

Quasi-implication algebras, Part I: Elementary theory

Abstract: Quasi-implication algebras (QIA's) are intended to generalize orthomodular lattices (OML's) in the same way that implication algebras (J. C. Abbott) generalize Boolean lattices. A QIA is defined to be a setQ together with a binary operation → satisfying the following conditions (a→b is denotedab). $$\left( {ab} \right)a = a$$ (Q1) $$\left( {ab} \right)\left( {ac} \right) = \left( {ba} \right)\left( {bc} \right)$$ (Q2) $$\left( {\left( {ab} \right)\left( {ba} \right)} \right)a = \left( {\left( {ba} \right)\left( {ab} \right)} \right)b$$ (Q3)

Topologies on products of partially ordered sets III: Order convergence and order topology

Abstract: This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are: Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.

Quasi-implication algebras, Part II: Structure theory

Abstract: The chief purpose of this paper is to show that the theory of quasi-implication algebras (QIA's) provides a complete characterization of the quasi-implication operation, as defined on orthomodular lattices (OML's) (on any OML,a→b=a⊥⋁(a⋀b)). This is accomplished by showing that every QIA can be embedded into an OML preserving quasi-implication. Besides yielding completeness, the embedding theorem provides the canonical class of QIA's, as follows. First of all, since QIA's are equationally defined, the class of QIA's is closed under the formation of subalgebras and homomorphic images. An orthomodular QIA is defined to be a QIA which is a quasi-implicational subalgebra of an OML. Together with the fact that QIA's are closed under the formation of homomorphic images, the embedding theorem entails that the class of orthomodular QIA's is canonical; specifically, every QIA is isomorphic to an orthomodular QIA.

A complete logic forn-permutable congruence lattices

Abstract: A family of logical systems, which may be regarded as extending equational logic, is studied. The equationsf=g of equational logic are generalized to congruence equivalence formulasf≡g (modx), wheref andg are terms interpreted as elements of an algebraV of some specified type. and termx is interpreted as a member of ann-permutable lattice of congruences forV. Formal concepts of proof and derivability from systems of hypotheses are developed. These proofs, like those of equational logic. require only finite algebraic processes, without manipulation of logical quantifiers or connectives. The logical systems are shown to be correct and complete: a well-formed statement is derivable from a system of hypotheses if and only if it is valid in all models of these hypotheses.

Hyperidentities of lattices and semilattices

Abstract: Following W. Taylor we define a hyperidentity ∈ to be formally the same as an identity (e.g.,F(G(x, y, z), G(x, y, z))=G(x, y, z)). However, a varietyV is said to satisfy a hyperidentity ∈, if whenever the operation symbols of ∈ are replaced by any choice of polynomials (appropriate forV) of the same arity as the corresponding operation symbols of ∈, then the resulting identity holds inV in the usual sense. For example, if a varietyV of type with operation symbols ∨ and ∧ satisfies the hyperidentity given above, then substituting the polynomial (x∨y)∨z for the symbolG, and the polynomialx∧y forF, we see thatV must in particular satisfy the identity ((x∨y)∨z)∧((x∨y)∨z)=((x∨y)∨z). The set of all hyperidentities satisfied by a varietyV, will be denoted byH(V). We shall letHm(V) be the set of all hyperidentities hoiding inV with operation symbols of arity at mostm, andHn(V) will denote the set of all hyperidentities ofV with at mostn distinct variables. In this paper we shall show that ifV is a nontrivial variety of lattices or the variety of all semilattices, then for any integersm andn, there exists a hyperidentity ∈ such that ∈ holds inV, and ∈ is not a consequence ofHm(V)∪Hn(V). From this it is deduced that the hyperidentities ofV are not finitely based, partly soling a problem of Taylor [7, Problem 3].

Molecules of a partially ordered set

Abstract: It is shown that in a general partially ordered set (P, ≤) the notion of a molecule (ie, a nonzero elementm ofP such that any two nonzero elements ofP which are ≤m have a nonzero lower bound) is in close analogy to the notion of an atom in a Boolean ring